The Roots of the Equation: A Comprehensive Analysis

Consider the equation involving the functions fx sin x and gx x/100. The objective is to determine the number of roots of the equation sin x x/100 over a specified interval. This analysis will utilize the properties of odd functions and the Intermediate Value Theorem to provide a detailed solution.

Introduction to Odd Functions and Their Properties

Both fx sin x and gx x/100 are odd functions. This implies that for any input x, sin(-x) -sin(x) and -x/100 -x/100. This property is crucial in determining the number of roots because if x0 is a root, then -x0 is also a root, providing symmetry around the origin.

Finding the Interval for Solutions

Due to the amplitude of the sine function being 1, we can restrict our attention to solutions where gx x/100 ≤ 1. This means we only need to consider the interval [0, 100], as sin x oscillates between -1 and 1.

Analysis of the Intersection Points

The function gx x/100 is always positive for x > 0. Meanwhile, fx sin x has a period of 2π. By examining each interval of the form [2kπ, 2kπ 2π], we can determine the number of intersection points.

Intersection Points within One Period

Within each half-period interval [0, π], [2π, 3π], and so on, the line gx x/100 intersects the sine curve at two points. This number of intersections can be shown officially with the Intermediate Value Theorem. To illustrate, consider the interval [0, π]. If we evaluate sin(π/2) 1 and sin(π) 0, and note that π/2 and π > 100/2, we can use the Intermediate Value Theorem to confirm the existence of two roots in this interval.

Application of the Intermediate Value Theorem

Let's consider the interval [0, 100] and apply the Intermediate Value Theorem to show that there are intersections. Since 100/2π ≈ 15.9, this means there are 15 full periods of sin x within this interval. Each full period contributes two intersection points, totaling 30 intersection points from these full periods. The remaining partial periods will also provide additional intersection points, thus leading to a total of 62 points of intersection.

Counting the Roots Symmetrically

By odd symmetry, for every root in [0, 100], there is a corresponding root in [-100, 0]. Therefore, the total number of roots is twice the number of roots in one half-interval, minus the root at the origin, which is counted twice.

Conclusion and Final Count

In summary, there are 32 solutions in the interval [0, 100] and 32 solutions in the interval [-100, 0]. Since the origin x 0 is included in both intervals, the total number of solutions is 2 * 32 - 1 63.

Key Observations

Odd functions: fx sin x and gx x/100 Intermediate Value Theorem: used to confirm the existence of root intersections Periodicity: the periodic nature of the sine function contributes to the number of intersection points Odd symmetry: utilized to count the total number of roots

This comprehensive analysis provides a clear method for determining the number of roots of the given equation, leveraging the properties of odd functions and the Intermediate Value Theorem.